Theorem conncompss 23551

Description: The connected component containing 𝐴 is a superset of any other connected set containing 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.)

Hypothesis

Ref	Expression
conncomp.2	⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}

Assertion

Ref	Expression
conncompss	⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ 𝑆)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋

Allowed substitution hints:   𝑆(𝑥)   𝑇(𝑥)

Proof of Theorem conncompss

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1152	. . . . 5 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ 𝑋)
2		conntop 23535	. . . . . . 7 ⊢ ((𝐽 ↾t 𝑇) ∈ Conn → (𝐽 ↾t 𝑇) ∈ Top)
3	2	3ad2ant3 1151	. . . . . 6 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → (𝐽 ↾t 𝑇) ∈ Top)
4		restrcl 23275	. . . . . . 7 ⊢ ((𝐽 ↾t 𝑇) ∈ Top → (𝐽 ∈ V ∧ 𝑇 ∈ V))
5	4	simprd 500	. . . . . 6 ⊢ ((𝐽 ↾t 𝑇) ∈ Top → 𝑇 ∈ V)
6		elpwg 4561	. . . . . 6 ⊢ (𝑇 ∈ V → (𝑇 ∈ 𝒫 𝑋 ↔ 𝑇 ⊆ 𝑋))
7	3, 5, 6	3syl 19	. . . . 5 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → (𝑇 ∈ 𝒫 𝑋 ↔ 𝑇 ⊆ 𝑋))
8	1, 7	mpbird 260	. . . 4 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ∈ 𝒫 𝑋)
9		3simpc 1166	. . . 4 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → (𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn))
10		eleq2 2854	. . . . . 6 ⊢ (𝑦 = 𝑇 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑇))
11		oveq2 7408	. . . . . . 7 ⊢ (𝑦 = 𝑇 → (𝐽 ↾t 𝑦) = (𝐽 ↾t 𝑇))
12	11	eleq1d 2850	. . . . . 6 ⊢ (𝑦 = 𝑇 → ((𝐽 ↾t 𝑦) ∈ Conn ↔ (𝐽 ↾t 𝑇) ∈ Conn))
13	10, 12	anbi12d 643	. . . . 5 ⊢ (𝑦 = 𝑇 → ((𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn) ↔ (𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn)))
14		eleq2 2854	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
15		oveq2 7408	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐽 ↾t 𝑥) = (𝐽 ↾t 𝑦))
16	15	eleq1d 2850	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐽 ↾t 𝑥) ∈ Conn ↔ (𝐽 ↾t 𝑦) ∈ Conn))
17	14, 16	anbi12d 643	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn) ↔ (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn)))
18	17	cbvrabv 3427	. . . . 5 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} = {𝑦 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn)}
19	13, 18	elrab2 3657	. . . 4 ⊢ (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ↔ (𝑇 ∈ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn)))
20	8, 9, 19	sylanbrc 594	. . 3 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
21		elssuni 4900	. . 3 ⊢ (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} → 𝑇 ⊆ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
22	20, 21	syl 18	. 2 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
23		conncomp.2	. 2 ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}
24	22, 23	sseqtrrdi 3980	1 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Conn) → 𝑇 ⊆ 𝑆)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  {crab 3417  Vcvv 3457   ⊆ wss 3907  𝒫 cpw 4558  ∪ cuni 4868  (class class class)co 7400   ↾_t crest 17463  Topctop 23011  Conncconn 23529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-rest 17465  df-top 23012  df-conn 23530

This theorem is referenced by:  conncompcld  23552  tgpconncompeqg  24230  tgpconncomp  24231